Constant of integration (Was Re: sqrt(x)*sqrt(x))

"Rich Hennessy" <richhen2008 at gmail.com> writes:

> Well, you are right.  I am the same way.  I don?t know Lisp so I would
> have a very hard time fixing anything that I see as wrong with Maxima.
> I can extend it with Maxima code, but that is as good as I can do.  I
> would like to be taught Lisp.  You can?t read minds, programming
> languages are invented, so you need someone already trained by the
> original writers of Lisp or one of their disciples to translate for
> you until you can understand on your own. Then you become a disciple
> too and can train others and so on . . .

The wonderful news is that people write books. And you can read them,
gaining knowledge. It's a very exciting experience. Sadly, I never had a
9th-Dan lisp master as a personal trainer, but I like to think I learnt
the language anyway.

>
> As far as the + C is concerned.  Actually I was not really criticizing
> the + C but the thread that has been going on forever it seems about
> sqrt(x^2) = abs(x).  I think I should say I like the absolute value
> function, some people don't.  They are not right, they just don't like
> it.  Maybe because it's not invented by them I guess.

The reason Dr. Fateman and others are talking about the identity

  sqrt(x^2) = abs(x)

is that there are two definitions of sqrt. The one that you are using is
that

  "sqrt(x) is the positive real number y such that y^2 = x".

The problem is that this doesn't make any sense over the complex
numbers. How can you extend the definition you want to the complex
plane? Note that you do have to do this, since many algorithms in Maxima
use complex numbers internally, even if the inputs and outputs are real.

The other option is that you say "sqrt(x) is the pair of numbers whose
squares are x" (well, it might be a singleton if x=0). But then the
identity above is nonsense. Even if you say that abs should apply to
each element of the set,

  sqrt(2^2) = sqrt(4) = {-2, 2} != {2} = abs (2)

Have you any good ideas for how to deal with this?


Rupert

P.S. Incidentally, please please don't top-post. It makes it a lot more
     effort to read your replies. See
     http://daringfireball.net/2007/07/on_top
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